The Physics of Music and Color
Leon Gunther
The Physics of Music
and Color
123
Leon Gunther
Department of Physics and Astronomy
Tufts University
Medford, Massachusetts
USA
[email protected]
ISBN 978-1-4614-0556-6
e-ISBN 978-1-4614-0557-3
DOI 10.1007/978-1-4614-0557-3
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011934793
© Springer Science+Business Media, LLC 2012
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Dedicated to my mother, Esther (Weiss)
Gunther Wand, who nurtured me with
a deep appreciation of music and the beauty
of nature, and to my wife, Joelle (Cotter)
Gunther, who sustains me with her love
and wisdom
Preface
This textbook has its roots in a course that was first given by Gary Goldstein and
me at Tufts University in 1971. Both of us are theoretical physicists, with Gary
focusing on the study of elementary particles and me focusing on condensed matter
physics, which is the study of the fundamental behavior of various types of matter
– superconductors, magnets, fluids, among many others. However, in addition, we
both have a great love and appreciation for the arts. This love is fortunately also
manifested in our involvement therein: Gary has been seriously devoted to oil
painting. I have played the violin since I was seven and played in many community
orchestras. I am also the founder and director of a chorus. Finally, I am fortunate to
have a brother, Perry Gunther, who is a sculptor and my inspiration and mentor in
the fine arts.
It is common to have a course on either the Physics of Music or the Physics
of Color. Numerous textbooks exist, many of which are outstanding. Why did we
choose to develop a course on both music and color? There are a number of reasons:
1. The basic underlying physical principles of the two subjects overlap greatly
because both music and color are manifestations of wave phenomena. In
particular, commonalities exist with respect to the production, transmission, and
detection of sound and light. Our decision to include both music and color
was partly due to the fact that some wave phenomena are relatively easy to
demonstrate for sound but not for light; they are experienced in every day life.
Examples include diffraction and the Doppler effect. Thus, the study of sound
helps us understand light. On the other hand, there are some wave phenomena
– common to both sound and light – that are more easily observed for light.
An example is refraction, wherein a beam of light is traveling through air and is
incident upon a surface of glass. Refraction causes the beam to bend upon passing
into the glass. Refraction is the basis for the operation of eyeglasses. And finally,
there are wave phenomena that are easily observable for both sound and light.
Interference is an example.
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Preface
Two stereo loudspeakers emitting a sound at the same single frequency
produce dead (silent) regions within a room as a result of the interference
between the two sound waves produced by the two loudspeakers; the colors
observed on the CDs of the photo in the frontispiece are a result of the
interference of light reflected from the grooves within the CDs.
2. The production of music and color involves physical systems, whose behavior
depends upon a common set of physical principles. They include vibrating
mechanical systems (such as the strings of the violin or the drum, vibrating
columns of air in wind instruments and the organ), electromagnetic waves such
as light, the rods and cones of the eye, and the atom. All manifest the existence of
modes and the phenomena of excitation, resonance, energy storage and transfer,
and attenuation.
CDs “produce” sound through a series of processes that involve many distinct
physical phenomena. First, the CD modulates a laser beam that excites an
electronic device into producing an electrical signal. The laser light itself is a
manifestation of electric and magnetic fields. The electrical signal is used to
cause the cone of a loudspeaker to vibrate and produce the motion in air that
is none other than the sound wave that we hear.
3. The course that led to the writing of this book offers us the opportunity to study a
major fraction of the basic principles of physics, with an added important feature:
Traditionally, introductory physics courses are organized so that basic principles
are introduced first and are then applied wherever possible. This course, on the
other hand, is based on a motivational approach: Because of the ease of observing
most phenomena that is afforded by including both light and sound, we are able
to introduce the vast majority of topics using class demonstrations.
We challenge ourselves by calling for a physical basis for what we observe.
We turn to basic principles as a means of understanding the phenomena. A study
of both subjects involves pretty nearly the entire gamut of the fundamental laws
of classical as well as modern physics. (The main excluded areas are nuclear and
particle physics and relativity.)
Ultimately, our approach helps us appreciate a central cornerstone of physics – to
uncover a minimal set of concepts and laws that is adequate to describe and account
for all physical observations. Simplification is the motto. We learn to appreciate how
it is that because the laws of physics weave an intricate, vast web among physical
phenomena, physics (and science generally) has attained its stature of reflecting
what some people refer to as “truth” and, much more significantly, of having an
extraordinarily high level of dependability.
The prerequisites for the associated course are elementary algebra and a familiarity with the trigonometric functions. The only material in the textbook that
requires a higher level of mathematics is the appendix on the Transformation of
Color Matching Functions (Appendix I) from one set of primaries to another –
the analysis requires a good understanding of matrices. I have never included this
appendix in my course; it is available for those who might be interested in it.
The level of the textbook is such as to produce questions as to whether a student
Preface
ix
without inclinations to major in the sciences can handle the material. It has been my
experience in teaching the associated course at Tufts University for over 35 years,
that very few such students have failed to do well in the course. In the Fall, 2009
semester, in particular, the 15 students who took the course were all majoring in the
Arts, Humanities, or Social Sciences or as yet had not declared a major. The average
score on the Final Exam was a respectable 73%, with a range from 61% to 94%.
When I have taught the course using this textbook, I have often had to omit the
section on Polarized Light for want of time. Sections that can be skipped without
loss of continuity for the remaining material are marked with a double asterisk (**).
Note on problems and questions: Whether you are reading this book in connection with a course you are taking or reading it on your own, I strongly urge
you to take the questions and problems in the book very seriously. To test your
understanding and to measure your level of understanding, you have to do problems.
In all my more than 50 years of studying physics, I have never truly appreciated a
new subject without doing problems.
There are many fine books already available that cover either the physics of sound
and music or the physics of light and color. Some of these books go into great depth
about a number of the subjects, way beyond the depth of this book. For example,
you will not find details on the complex behavior of musical instruments in this
book. The book by Arthur Benade, listed in the Appendix of references D, is a great
resource on this subject, even though it is quite dated. And, you will not find indepth coverage of the incredibly rich range of light and color phenomena that is
treated in the wonderful book by Williamson and Cummins. Their section on oil
paint is outstanding. Instead, you should look on this book as a resource for gaining
an in-depth understanding of the relevant concepts and learning to make simple
calculations that will help you test hypotheses for understanding phenomena that
are not covered in this book. You will be able to read other books and articles on
the web empowered with an understanding that will help you appreciate the content.
One of the problems raging today (2011) is the proliferation of information. Ah yes,
you can look up on the Web any topic in this book. Unfortunately, a huge fraction of
the information is incorrect or unreliable.1 How can you judge what you read? The
1
Recently, the SHARP Corporation announced that it was going to make available a color monitor
and TV that has four primary colors among the color pixels, in contrast to the three primaries
currently used. As a result, it claimed that the number of colors available would approach one
trillion. (See their website: http://www.sharpusa.com/AboutSharp/NewsAndEvents/PressReleases/
2010/January/2010 01 06 Booth Overview.aspx) Yet you will learn in Chap. 14 that human vision
can differentiate only about ten million colors. Therefore, even if the Sharp monitor were able to
produce one trillion colors, viewers would not be able to benefit from this great technology. We can
still ask what can possibly be the gain in adding a yellow primary? Is their chosen color yellow for
the fourth primary the best one to choose to improve our color vision? See Chap. 14 for information
on this question. Websites abound dealing with the significance of Sharp’s new technology; this
book will help you analyze and judge what you read.
x
Preface
only solution is for you to accumulate knowledge and understanding of the basics
and to criticize what you read.2
Acknowledgements First and foremost I am indebted to Gary Goldstein, who was
a co-developer of the original course on The Physics of Music and Color. Gary’s
contributions in teaching a number of the subjects in a clear way were invaluable.
Most noteworthy were his ideas for teaching color theory. I am grateful to my
daughter, Rachel Gunther, for producing the first word-processed draft of the book.
I am deeply indebted to Stephan Richter, one of my graduate students, who was a
driving force and indefatigable in producing a Latex copy of the book, worked over
numerous figures, and is responsible for the layout of the book. I had a number of
teaching assistants over the years who made very valuable contributions in teaching
the course, most notably Stephan Richter and Rebecca Batorsky. Both Stephan
and Rebecca are gifted teachers and frequently shared productive advice for me.
My long time friend and violin teacher, Wolfgang Schocken, was a well-known
teacher of the violin. He was also extremely knowledgeable about the numerical
issues involved in intonation, which he shared with me. In spite of my familiarity
with resonances and overtones of a vibrating string, it was he who taught me to
listen carefully to the resonant vibration of unbowed strings to vastly improve my
intonation. My son, Avi Gunther, who got his Bachelor’s degree from the Berklee
College of Music in Boston with a major in Music Production and Engineering, was
often extremely valuable in advising me on many aspects of music and on sound
production.
I benefitted greatly from two readers of this book: The first reader was my
personal opthalmologist, Dr. Paul Vinger, who pointed out numerous typos and
provided me with questions that he suggested be addressed in the book. My second
reader was a student of mine, Bryce Meyer, who did an incredibly dedicated
job reading carefully through the book – finding typos and making countless
suggestions for improving the clarity of various passages in the text. Bryce also
helped me with some figures.
Many individuals have helped me in one way or another toward the writing
of this book. I list the following with apologies those who should be here but
are omitted: Paavo Alku, Anandajoti Bhikku, Bruce Boghosian, Andrew Bregman,
Andrew Clarke, David Copenhagen, Tom Cornsweet, Russ Dewey, Marcia Evans,
Oliver Knill, Paul Lehrman, Ken Lang, Jay Neitz, Donna Nicol, Ken Olum, Charles
Poynton, Jeffrey Rabin, Brian Roberts, Judith Ross, Eberhard Sengpiel, George
Smith, and Raymond Soneira. This book would not have been published were it not
for the strong support and help of my editors, Christopher Coughlin and HoYing
2
What applies to information on science applies to all subjects. If you are given a multitude of
conflicting expert opinions on a subject, you will tend to choose one expert who is closest to your
point of view or you will want to throw all the sources out the window with the conclusion that
reliable information not only cannot be found but has no meaning. The fascinating book by Neil
Postman – Amusing Ourselves to Death [Penguin Books, N.Y, 1986] – discusses some related
problems connected with this proliferation of information.
Preface
xi
Fan. I want to pay special attention to Kaća Bradonjić, who produced tens of figures
with great finesse, especially those in Chap. 5 that are based on my crude hand
drawings.
This book has been a work in progress for more than 35 years. It has had many
drafts. I need to share with you my deep appreciation for my loving wife, Joelle,
for supporting me in this effort. Whenever I needed encouragement to sustain my
spirits and energy, Joelle was there for me.
Questions Discussed in This Book
1. Why is the sky blue and the setting sun red?
2. How does the rainbow get its colors?
3. How is it that all light is a mixture of the colors of the rainbow? Yet the color
brown is not simply a mixture of these colors?
4. How is it that sound can bend around corners?
5. Does light bend around corners?
6. What simple mathematical relationships form the bases of the musical scales of
most of the world’s cultures? Are these relationships unique?
7. Are there three primary colors?
8. What are the colors white, black, gray, and brown?
9. How is the eye like a camera?
10. How is it that the ear can perceive two distinct musical tones, yet the eye
perceives a mixture of two colors as a single color?
11. How can we get color from purely black and white images?
12. How does the brain determine the direction of a source of sound?
13. What is noise?
14. Why does the trumpet sound different from the violin?
15. What is a mirage?
16. Why do stars seem to twinkle?
17. How do color prints, color slides, and color TV work?
18. Can a soprano really break glass?
19. Why does a flutist have to retune his or her flute a while after having begun
playing?
20. How is sound transmitted electrically?
21. How does the ear provide us with a sense of pitch?
22. Can a fish hear a fisherman talking?
23. Why do some automobiles rattle at a speed of about 55 mph?
24. How can we hear sounds which are not in the air? How is this phenomenon
related to the blue color of the ocean?
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Questions Discussed in This Book
25. How can a person hear a clock ticking at a frequency of one tick per second,
while it is said that the lowest frequency that can be heard is about 20 cycles
per second?
26. How can we estimate the speed of an overhead propeller-driven airplane from
the sound it emits?
27. How does the vibrato of a violin help improve our perception of consonance
among groups of notes?
28. Why does it become more difficult to perceive a sense of pitch as we play ever
lower-pitched notes on a piano?
Contents
1
Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1
The Legend of the Huang Chung . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1
7
2
The Vibrating String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1
Waves Along a Stretched String.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2
A Finite String Can Generate Music! . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3
Pitch, Loudness, and Timbre . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4
The Relation Between Frequency and Pitch . . .. . . . . . . . . . . . . . . . . . . .
2.5
The Wave Motion of a Stretched Rope . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6
Modes of Vibration and Harmonics .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.7
The Sine Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.8
The Simple Harmonic Oscillator . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.8.1
The Vibration Frequency of a Simple
Harmonic Oscillator . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.9
Traveling Sine Waves . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.9.1
Applications .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.10 Modes of Vibration: Spatial Structure . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.11 The Wave Velocity of a Vibrating String . . . . . .. . . . . . . . . . . . . . . . . . . .
2.11.1 Application of the Above Relations to the Piano .. . . . . .
2.12 The Connection Between an SHO and a Vibrating String . . . . . . . .
2.13 Stiffness of a String .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.14 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.15 General Vibrations of a String: Fourier’s Theorem . . . . . . . . . . . . . . .
2.15.1 Frequency of a Wave with Missing Fundamental . . . . . .
2.16 Periodic Waves and Timbre . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.17 An Application of Fourier’s Theorem to Resonance
Between Strings .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.18 A Standing Wave as a Sum of Traveling Waves . . . . . . . . . . . . . . . . . . .
2.19 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.20 Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.21 Problems for Chap. 2.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11
11
13
16
17
18
20
23
26
28
29
31
32
34
37
38
41
43
45
51
52
52
55
55
57
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3
4
Contents
The Vibrating Air Column . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1
The Air of Our Atmosphere . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1.1
Generating a Sound Pulse . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1.2
Digression on Pushing a Block of Wood . . . . . . . . . . . . . . .
3.2
The Nature of Sound Waves in Air.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3
Characterizing a Sound Wave . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4
Visualizing a Sound Wave . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5
The Velocity of Sound . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.1
Temperature Dependence of Speed
of Sound in Air. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6
Standing Waves in an Air Column . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.1
Standing Waves in a Closed Pipe . . .. . . . . . . . . . . . . . . . . . . .
3.6.2
End Correction for Modes in a Pipe . . . . . . . . . . . . . . . . . . . .
3.7
Magic in a Cup of Cocoa . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.8
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.9
Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.10 Problems for Chap. 3.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.10.1 Derivation of the Helmholtz Formula .. . . . . . . . . . . . . . . . . .
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66
67
67
69
70
71
Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1
Forms of Energy and Energy Conservation.. . .. . . . . . . . . . . . . . . . . . . .
4.1.1
Fundamental Forms of Energy . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1.2
“Derived” Forms of Energy . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1.3
The Energy of Cheerios . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2
The Principle of Conservation of Energy, Work, and Heat . . . . . . .
4.3
Energy of Vibrating Systems . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3.1
The Simple Harmonic Oscillator . . . .. . . . . . . . . . . . . . . . . . . .
4.3.2
Energy in a Vibrating String.. . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3.3
Energy in a Sound Wave. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4
Power .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.5
Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.6
Intensity of a Point Source . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.7
Sound Level and the Decibel System . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.7.1
Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.7.2
Sound Level .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.7.3
From Sound Level to Intensity . . . . . .. . . . . . . . . . . . . . . . . . . .
4.8
Attenuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.8.1
Attenuation in Time.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.8.2
Resonance in the Presence of Attenuation .. . . . . . . . . . . . .
4.8.3
Attenuation of Travelling Waves:
Attenuation in Space .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.9
Reverberation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.10 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.11 Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.12 Problems for Chap. 4.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
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89
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95
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96
98
99
99
101
103
105
105
107
108
110
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Contents
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5
Electricity, Magnetism, and Electromagnetic Waves . . . . . . . . . . . . . . . . . . .
5.1
The Fundamental Forces of Nature . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2
The Electric Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3
Electric Currents in Metal Wires . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4
The Magnetic Force.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.5
Magnetic Forces Characterized . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.6
Is There a Connection Between Electricity and Magnetism? .. . . .
5.6.1
Action–Reaction Law and Force of Magnet
on Current-Carrying Wire . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.7
The Loudspeaker .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.8
The Buzzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.9
The Electric Motor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.10 Force Between Two Wires Carrying an Electric Current . . . . . . . . .
5.11 The Electromagnetic Force and Michael Faraday .. . . . . . . . . . . . . . . .
5.12 Applications of Faraday’s EMF . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.13 A Final “Twist” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.14 Action-at-a-Distance and Faraday’s Fields . . . .. . . . . . . . . . . . . . . . . . . .
5.15 The Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.16 The Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.17 Magnetic Force on a Moving Charge . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.18 Force Between Two Parallel Wires Carrying Currents .. . . . . . . . . . .
5.19 Generalized Faraday’s Law.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.20 What Do Induced Electric Field Lines Look Like? . . . . . . . . . . . . . . .
5.21 Lenz’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.22 The Guitar Pickup.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.23 Maxwell’s Displacement Current . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.24 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.25 What Is the Medium for Electromagnetic Waves? . . . . . . . . . . . . . . . .
5.26 The Sources of Electromagnetic Waves . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.27 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.28 Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.29 Problems for Chap. 5.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
127
127
129
130
131
133
135
The Atom as a Source of Light . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1
Atomic Spectra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2
The Hydrogen Spectrum of Visible Lines . . . . .. . . . . . . . . . . . . . . . . . . .
6.3
The Bohr Theory of the Hydrogen Atom .. . . . .. . . . . . . . . . . . . . . . . . . .
6.4
Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5
Complex Scenarios of Absorption and Emission.. . . . . . . . . . . . . . . . .
6.5.1
Rayleigh Scattering . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5.2
Resonance Fluorescence .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5.3
General Fluorescence .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5.4
Stimulated Emission . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.6
Is Light a Stream of Photons or a Wave? . . . . . .. . . . . . . . . . . . . . . . . . . .
6.7
The Connection Between Temperature and Frequency . . . . . . . . . . .
179
179
181
184
190
195
196
196
196
197
199
200
6
138
141
141
142
143
143
147
148
149
150
154
157
158
158
163
164
166
167
169
174
175
177
178
178
xviii
Contents
6.8
6.9
6.10
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 202
Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 202
Problems for Chap. 6.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 203
7
The Principle of Superposition .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1
The Wave Produced by Colliding Pulses . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2
Superposition of Two Sine Waves of the Same Frequency . . . . . . .
7.3
Two Source Interference in Space.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3.1
Sound Level with Many Sources .. . .. . . . . . . . . . . . . . . . . . . .
7.3.2
Photons and Two-Slit Interference ... . . . . . . . . . . . . . . . . . . .
7.4
Many-Source Interference .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.1
Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.2
Diffraction Through a Mesh . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.3
X-ray Diffraction of Crystals . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5
Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.6
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.7
Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.8
Problems for Chap. 7.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
205
205
207
209
216
216
217
217
218
220
221
224
224
225
8
Propagation Phenomena .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1
Diffraction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1.1
Scattering of Waves and Diffraction . . . . . . . . . . . . . . . . . . . .
8.1.2
Why Is the Sky Blue? . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2
Reflection.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2.1
A Complex Surface: A Sand Particle . . . . . . . . . . . . . . . . . . .
8.3
Reflection and Reflectance . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3.1
The Reflectance for a Light Wave. . .. . . . . . . . . . . . . . . . . . . .
8.3.2
The Reflectance for a Sound Wave. .. . . . . . . . . . . . . . . . . . . .
8.4
Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.5
Total Internal Reflection .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.6
The Wave Theory of Refraction .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.7
Application to Mirages . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.8
The Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.9
Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.9.1
Effect of Dispersion on a Prism . . . . .. . . . . . . . . . . . . . . . . . . .
8.9.2
Effect of Dispersion on Fiber Optics Communication .
8.10 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.1 The Converging Lens . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.2 Lens Aberrations .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.3 Image Produced by a Converging Lens .. . . . . . . . . . . . . . . .
8.10.4 Magnification . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.5 Reversibility of Rays: Interchange
of Object and Image . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.6 The Diverging Lens . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.10.7 Determining the Focal Length
of a Diverging Lens .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
231
231
238
240
241
244
245
246
248
249
251
252
255
256
257
257
258
259
259
260
264
266
269
269
271
Contents
8.11
8.12
8.13
8.14
8.15
9
xix
The Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.11.1 Doppler Effect for Waves in a Medium .. . . . . . . . . . . . . . . .
8.11.2 Doppler Effect for Electromagnetic Waves
in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.11.3 Applications of the Doppler Effect... . . . . . . . . . . . . . . . . . . .
Polarized Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.1 How Can We Obtain a Beam of Polarized Light? .. . . . .
8.12.2 Series of Polarizers . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.3 Ideal vs. Real Polarizers . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.4 Sample Problems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.5 Partial Polarization of Reflected Light .. . . . . . . . . . . . . . . . .
8.12.6 The Polarization of Scattering Light .. . . . . . . . . . . . . . . . . . .
8.12.7 The Polarizer Eyes of Bees . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.8 Using Polarization of EM Radiation
in the Study of the Big Bang . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.9 Optical Activity . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.12.10 Our Chiral Biosphere . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Important Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Questions and Problems for Chap. 8 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.1
Broad Outline of the Conversion Process . . . . . .. . . . . . . . . . . . . . . . . . . .
9.2
The Auditory Canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.3
The Eardrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.4
The Ossicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5
Improving on the Impedance Mismatch: Details . . . . . . . . . . . . . . .
9.6
The Cochlea .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.7
Pitch Discrimination .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.7.1
Some Mathematical Details on Pitch vs. the
Peak of the Envelope . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.7.2
Mach’s Law of Simultaneous Contrast in Vision.. . . . . .
9.7.3
Rhythm Theory of Pitch Perception . . . . . . . . . . . . . . . . . . . .
9.8
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.9
Problems for Chap. 9.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10 Psychoacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.1 Equal Loudness Curves .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.2 The “Sone Scale” of Expressing Loudness .. . . . . . . . . . . . . . . . . . . .
10.3 Loudness from Many Sources .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.4 Combination Tones and the Nonlinear Response of the Cochlea.
10.5 The Blue Color of the Sea and Its Connection
with Combination Tones . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.6 Duration of a Note and Pitch Discrimination ... . . . . . . . . . . . . . . . . . . .
10.7 Fusion of Harmonics: A Marvel of Auditory Processing . . . . . . . . .
10.7.1 Mathematica File . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
272
273
277
278
280
281
282
283
284
285
286
286
287
287
291
293
293
294
305
306
310
310
311
313
315
318
319
322
322
324
325
326
327
329
331
334
335
341
342
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